fined Swelling-induced deformations: a materials-de transition from macroscale to microscale deformations

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Soft Matter
COMMUNICATION
Cite this: Soft Matter, 2013, 9, 5524
Swelling-induced deformations: a materials-defined
transition from macroscale to microscale deformations
Anupam Pandey and Douglas P. Holmes*
Received 11th January 2013
Accepted 3rd May 2013
DOI: 10.1039/c3sm00135k
www.rsc.org/softmatter
Swelling-induced deformations are common in many biological and
industrial environments, and the shapes and patterns that emerge
can vary across many length scales. Here we present an experimental
study of a transition between macroscopic structural bending and
microscopic surface creasing in elastomeric beams swollen nonhomogeneously with favorable solvents. We show that this transition is dictated by the materials and geometry of the system, and we
develop a simple scaling model based on competition between
bending and swelling energies that predicts if a given solvent droplet
would deform a polymeric structure macroscopically or microscopically. We demonstrate how proper tuning of materials and geometry
can generate instabilities at multiple length scales in a single
structure.
Designing advanced materials to accommodate uid–structure
environments requires accurate control over the swellinginduced deformations of so mechanical structures. The
dynamics of osmotically driven movements within elastic
networks, and the interplay between a structure's geometry
and boundary conditions, play a crucial role in the morphology
of growing tissues and tumors,1,2 the shrinkage of mud3 and
moss,4 and the curling of cartilage,5 leaves,6–8 and pine cones.9
Various structural and surface instabilities can occur when a
favorable solvent is introduced to a dry gel. These include
buckling,10–12 creasing,13 and wrinkling instabilities,14–19 and
the curling of paper and rubber.21–24 In addition, porous thin
lms, such as fuel cell membranes,25,26 are highly susceptible
to swelling-induced delamination and buckling, which cripple
their functionality. For materials to adequately accommodate
different uid–structure environments there must be an
accurate understanding of both the global and surfaceconned deformations that can occur. In this paper, we
examine the transition between global structural deformations
and surface patterns that occur as a function of material
Virginia Tech, Engineering Science and Mechanics, 222 Norris Hall, Blacksburg, VA,
USA. E-mail: [email protected]; Tel: +1 540 231 7814
5524 | Soft Matter, 2013, 9, 5524–5528
geometry, uid–structure interaction, and the volume fraction
of the swelling uid.
Building on the idea that soil can be treated as a porous
elastic network,27 the consolidation of uid-saturated soil was
examined,28 and the mathematical framework for poroelasticity
that emerged29–31 remains the dominant theory for describing
the migration of uid within gels,32,33 tissues,34,35 and granular
media.31,36 The uid movement is dictated by the characteristic,
or poroelastic, time required for it to move through a network. If
the uid and network are incompressible, and the uid velocity
obeys Darcy's law,28 a diffusive-like time scale emerges: s L2D1, where L is the characteristic length scale, and D is the
diffusivity. The steady permeation of uid through rigid porous
networks, and the equilibrium swelling of crosslinked elastic
networks are well described by these diffusion dynamics, but at
early times the dynamic deformations of these so, swelling
elastomers can be highly nontrivial.
Material geometry, structural connement, and nonhomogenous exposure to uid will all add additional complications to the swelling-induced deformations of so mechanical structures. For instance, when the surface of a dry gel is
exposed to favorable solvent, how does it accommodate the
stress that develops? If a slender beam is exposed on one face to
a favorable solvent the uid will fully swell the beam at long
times, and it will reach a larger equilibrium length.32,37 At short
times, the outer surface of the beam is under more stress than
the bulk, and will elongate and bend into an arch.22–24,38,39
However, bending of the structure will only occur if the swelling
stress is greater than the stress required to bend the beam.
If the material is very thick, the resistance to bending will
generate a compressive stress on the surface, causing the
outermost layer of the structure to crease. Biot was rst to
predict the presence of surface instabilities on a homogeneous
elastic medium due to compressive forces.44 Predicting and
controlling these surface instabilities will impact the design of
exible electronics, so robotics, and microuidic devices.
While most of the available literature on wrinkling and creasing
deals with homogeneous swelling of conned gels submersed
This journal is ª The Royal Society of Chemistry 2013
Communication
in solvent13,14,40,41 or so substrates with stiff skins,15–17,19
understanding of the effect of elastomer geometry and uid
volume in case of the non-homogeneous swelling of gels is still
lacking. We present a simple model experiment for probing this
transition.
Fig. 1 illustrates the experimental procedure. We place a
favorable uid droplet of volume Vf on top surface of an elastomeric beam having undeformed length L. As the uid swells
the polymeric network, we measure the length of the beam, l(t),
and surface topography, l(t), as a function of time. The schematic in Fig. 1a shows the propagation of uid within the
crosslinked elastomer. The unswollen polymer coils within the
network have an equilibrium radius of gyration, which will
increase, i.e. swell, when the coil comes in contact with a solvent
that has favorable enthalpic interactions (Fig. 1a(i and ii)). A
balance of the enthalpic interactions between the polymer and
solvent and the entropic cost of stretching the chain dictate the
extent of swelling within a given material system.20 This local
expansion of the elastic network creates a structure analogous
to a bilayer, in which the top layer wants to be longer than the
bottom layer, and thus it bends (Fig. 1b). If the exural rigidity
of the beam is larger than the moment induced by swelling,
then the rigid structure will impose a compressive stress on the
Soft Matter
swollen layer, analogous to a gel that is constrained at its base
(Fig. 1c). If this stress creates a compressive strain that exceeds a
critical value, the surface will crease.
To study this transition between bending and creasing
experimentally, we prepared polydimethylsiloxane (PDMS)
(Dow Corning Sylgard 184) beams of various thicknesses
(1 mm, 2 mm, 3 mm, 4 mm, 5 mm) with a width b ¼ 2.5 mm
and length L ¼ 35 mm at a 10 : 1 (w/w) ratio of prepolymer-tocrosslinker. A droplet of uid was deposited at the middle of
the beam by a syringe pump (New Era Pump Systems Inc,
Model NE-1000), and its volume was varied from 1.29 mL to
5.96 mL by appropriate choice of syringe needle diameter
(Hamilton). As the beam swells, its structural deformation was
captured at 30 fps using Edmund-Optics GigE camera with a
Nikkor lens (35 mm f: 1–1.4), while its surface deformation
were recorded at 21 fps with a Edmund-Optics GigE camera
attached with an inverted Nikkor lens (35 mm f: 1–1.4) and a
macro extension tube (Vivitar N-AF 36 mm). Image analysis
was performed using ImageJ and Matlab to extract the
macroscopic beam elongation and microscopic crease spacing
as a function of time.
Fluids were chosen based on their affinity for swelling
PDMS,42 which is comprised by similarities in both solubility ds
and polarity m (Table 1). The solubility and polarity of a uid
within a network combine in a nontrivial way to swell the elastic
structure,42 therefore the measured strain at equilibrium for a
beam that is fully submersed in the uid, 3eq, is the metric we
use to differentiate swelling ability. Fig. 2a shows the swelling of
PDMS beams by ve different uids. Each beam is fully
submerged in a solvent and its elongation is measured as a
function of time. In this gure, we highlight the elongation at
short times (i.e. t # 100 s) to show that the beam initially
deforms faster than the diffusive dynamics that drive swelling at
long times,43 and that the rate of this early deformation differs
for various solvents. In particular, we note that polarity differences between solvent and network may play an important role
in the early time dynamics, with large deformations occurring
when |Dm| ¼ |Dms DmP| < 0.5.
To study the transition between macroscopic structural
deformation and surface deformation, we performed experiments with a nite volume of uid. When a droplet of volume Vf
is placed on the surface of a dry beam, a non-homogeneous
strain eld develops during swelling which propagates over
time. For thin beams, a macroscopic strain is measured from
the change in beam length, 3 ¼ l L/L. Fig. 2b(i) shows a graph
Table 1 Material solubility ds and polarity m obtained from Lee et al.,42 strain at
equilibrium measured experimentally
Fig. 1 Bending and creasing of beams swollen non-homogenously. (a) A schematic showing the propagation of fluid through crosslinked polymer network. (b)
A thin beam (h ¼ 1 mm, L ¼ 35 mm) is swollen by placing a diisopropylamine
droplet of volume 5.96 mL on its top surface. As the beam is fixed at both ends, it
bends out of plane to accommodate the expansion of the top surface. (c) A much
thicker beam (h ¼ 5 mm, L ¼ 35 mm) is swollen with the same solvent and
volume, but instead of bending, creases appear on the top surface.
This journal is ª The Royal Society of Chemistry 2013
Material
ds (cal1/2 cm3/2)
m (D)
3eq
PDMS
Diisopropylamine
Triethylamine
Hexanes
Toluene
Ethyl acetate
7.3
7.3
7.5
7.3
8.9
9.0
0.6–0.9
1.2
0.7
0.0
0.4
1.8
—
1.13
0.58
0.35
0.31
0.18
Soft Matter, 2013, 9, 5524–5528 | 5525
Soft Matter
Communication
Fig. 2 Swelling dynamics of PDMS at short times. (a) Submersed swelling strain
of PDMS beams by different solvents. Inset: different strain rates imparted by
different solvents at short times. (b) (i) Change in macroscopic strain over a period
of time when a diisopropylamine droplet of volume Vf ¼ 5.96 mL is placed on
beams of thicknesses 1 mm, 3 mm. (ii) Maximum strain developed in beams of
various thicknesses due to 5.96 mL droplet of all solvents. (iii) Maximum strain in 1
mm thick beam due to different droplet volume of all solvents.
of 3 as a function of time for two beams with different thicknesses that are exposed to 5.96 mL of diisopropylamine. This
strain grows until the imbibition front reaches the centerline of
the beam, and then gradually decreases as the uid continues to
propagate. At long times, the beam returns to a at state aer
the uid has evaporated from the network. We can quantify the
maximum macroscopic strain 3max that is achieved as a function
of beam thickness (Fig. 2b(ii)) and uid volume (Fig. 2b(iii)).
Above a critical thickness, a particular type and volume of uid
is unable to generate enough stress to create a macroscopic
deformation of the beam. We seek a reduced-order scaling
analysis that will describe the material and geometric contributions to this critical threshold.
In these swelling-induced deformations, there exists an
competition between the solvent's ability to swell the network,
and the structure's ability to resist deformation. Since the
swelling is non-homogenous, the strain variation through
the thickness of the beam may induce a bending deformation.
The rigidity of a structure can be quantied by the energy
required to bend it, which is:
ðL
Ub ¼
0
B 0
Eh3
ðq ðsÞÞ2 ds ;
2
12ð1 n2 Þ
(1)
Eh3
, h is the thickness of the beam, E is the
where B ¼
12ð1 n2 Þ
elastic modulus, n is Poisson's ratio, and q(s) is the angle along
5526 | Soft Matter, 2013, 9, 5524–5528
the arc-length. Since the time scale of swelling is short, and the
ratio of the uid to the solid elastic network is very small, we
assume a linear elastic behavior. If instead we allowed the
swollen structures to approach chemical equilibrium, it would
be necessary to consider a hyperelastic model where the elastic
constants depend on the swelling strain. With these assumptions, we expect an energy due to swelling will take a similar
ÐL
form, with Us ¼ 0 b, where b is a function of the elastic properties of the network, the interaction between the polymer and
solvent, and the amount of uid, i.e. b ¼ f (E, 3, Vf). If this takes
Ð
the form of a strain energy, we may write Us Vs s3dVs , where Vs
is the volume of the swollen region in the structure. With a
nite volume of uid, the maximum deformation will occur at
short times during the swelling process, when there is a high
concentration of solvent within a local region of the polymer
network. If this small, but highly concentrated region is analogous to a fully swollen structure, then at short times we may
assume that the strain imparted on these chains is on the order
of the equilibrium swelling strain 3 z 3eq, and the volume of the
swollen region will be conned to the volume of the uid Vs z
Vf. With these assumptions†, we expect the swelling energy to
scale as:
ð
Us ¼ s3eq dVf E3eq 2 Vf :
(2)
Vf
The swelling-induced deformation of these beams will be
dictated by the interplay between the bending and swelling
energies, such that no bending will occur when Ub Us. Hence
these competition between bending and swelling energy gives
rise to a length which we call elastoswelling length scale, ‘es:
‘es (3eq2Vf)1/3.
(3)
In the case of a transversely swollen beam, this length scale
denotes a critical thickness below which bending will occur.
In Fig. 3a, we plot the experimentally observed maximum
strain 3max as a function of beam thickness normalized by this
elastoswelling length, h/‘es. The critical threshold determined
from the reduced-order scaling prediction describes the onset
of macroscopic bending very well. We observed macroscopic
deformation with beams swollen by hexane and ethyl acetate
slightly above h/‘es ¼ 1, which may be attributed to an importance of |Dm| at short times. Further experimental studies that
focus on the impact of solvent–polymer polarity will be necessary to properly account for this phenomenon.
When h > ‘es, the swelling-induced stress will be insufficient
to macroscopically bend the structure, and the local expansion
of the top surface of the beam will be restricted by the bulk
PDMS network. Similar to gels that are conned to a rigid
substrate, the restriction against bending will impart a
compressive stress on the swelling region. If this compressive
stress imparts a strain greater than 3 ¼ 0.35, a localized surface
creasing instability will occur.44–46 Fig. 3b shows the development of surface creases as a thick beam (h ¼ 5 mm) swells with
diisopropylamine (Vf ¼ 5.96 mL) such that h/‘es z 1.22.
Randomly oriented surface-conned deformations appear
immediately aer swelling begins. Following the procedure
This journal is ª The Royal Society of Chemistry 2013
Communication
Soft Matter
Fig. 4 Bending and creasing. A structure with a variable thickness, such that it
exhibits both bending in the thin regions and creasing in the thick regions for the
same droplet size of solvent.
Fig. 3 Transition between global bending to surface creasing. (a) Maximum
strain in beams due to non-homogeneous swelling of different fluid droplet of
different volume, as a function of beam thickness normalized by elastoswelling
length. (b) Images of creases and their respective 2D FFT results. (c) Growth of
crease spacing with time.
described by Ebata et al.,47 we performed a 2D Fast Fourier
Transform (FFT) on the images to get the frequency and spacing
of these microstructures (Fig. 3a). At time zero, when no uid is
present, no structure is observed. On applying the uid droplet,
creases start to form across the surface, and peaks are evident
on the amplitude vs. frequency plot. These peaks come closer to
each other as the creases grow in length with time. Fig. 3b(ii)
conrms the expected relation from Tanaka et al.48 that the
characteristic crease spacing will scale as l t1/2. Our experimental results suggest that macroscopic deformation (bending)
will occur when the thickness is below the elastoswelling length
scale, i.e. h/‘es < 1, while microscopic deformations (creases) will
form when h/‘es > 1. In the region where h/‘es z 1, a coexistence
between both macroscale and microscopic deformations exist,
suggesting that although we were unable to observe creases
when h/‘es < 1, they may exist at very short times. Similarly, there
will likely be an upper limit of h/‘es, dictated by the surface
strain, in which no deformation occurs. Our experimental
approach makes accurately quantifying the local surface strains
difficult, so determining this upper threshold is beyond the
scope of this paper.
Finally, we demonstrate that both macroscopic structural
and microscopic surface deformation can be generated in the
same material by appropriately tuning the thickness. Fig. 4
shows a beam with a variable thickness along its length. In the
thin regions, macroscopic bending is observed and the surface
remains smooth, while in the thick region the surface develops
creases as it remains undeformed in macroscale. This
This journal is ª The Royal Society of Chemistry 2013
represents a potential mechanism for controlling shape change
and actuation across multiple length scales by generating
deformation simultaneously on the microscale and the
macroscale.
In summary, we have demonstrated a materials and geometry dened transition between a swelling-induced structural
deformation and surface deformation. We present a scaling
model that denes a critical elastoswelling length scale based on
a balance of uid–structure interactions, uid volume, and the
bending energy of the structure. Further experimental analyses
are necessary to conrm the extension of this scaling to other
geometries. Additionally, while our reduced-order model accurately captures the swelling-induced deformations in this paper,
a more thorough theoretical model is required to quantitatively
describe the scaling presented here. We anticipate that a better
understanding of the nature of swelling-induced deformations
of homogeneous materials will aid in designing and controlling
shape change across many structural geometries and length
scales. In particular, we provide a simple means for generating
both the surface patterns and global shape within a structure
using a given stimulus.
Acknowledgements
Funding for this work was provided by the National Science
Foundation (NSF CMMI-1300860) and the Department of
Engineering Science and Mechanics at Virginia Tech. The
authors acknowledge the helpful feedback from Dominic Vella
and Peter Stewart.
Notes and references
† The assumption that 3 z 3eq likely overestimates the local, short time strains
within the polymer coils, while assuming that Vs z Vf likely underestimates the
volume of the swollen region at the time of maximum deformation.
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